A264919 -id:A264919 - cp's OEIS frontend

A264918 Decimal expansion of constant z = Sum_{n>=1} {(3/2)^n} / 2^n, where {x} denotes the fractional part of x.

3, 9, 3, 1, 8, 8, 4, 7, 7, 0, 4, 9, 6, 4, 4, 3, 2, 4, 4, 9, 7, 2, 5, 8, 2, 1, 3, 1, 3, 8, 9, 0, 3, 8, 8, 5, 8, 5, 4, 8, 3, 9, 1, 4, 0, 7, 8, 8, 6, 6, 2, 8, 6, 9, 5, 3, 9, 2, 9, 3, 2, 4, 7, 5, 7, 5, 7, 8, 7, 7, 5, 8, 3, 3, 8, 9, 7, 4, 9, 8, 6, 6, 8, 1, 0, 9, 7, 6, 6, 6, 2, 0, 6, 1, 0, 1, 8, 5, 8, 8, 8, 0, 1, 3, 3, 3, 0, 0, 8, 0, 5, 9, 3, 2, 2, 6, 3, 1, 5, 3, 2, 6, 8, 0, 9, 0, 4, 7, 5, 0, 4, 9, 4, 2, 6, 6, 6, 1, 2, 1, 1, 4, 2, 4, 3, 3, 4, 9, 8, 4, 4, 3, 5, 8, 4, 7, 7, 5, 8, 5, 0, 6, 5, 5, 9, 3, 3, 7, 2, 5, 0, 9, 1, 4, 3, 2, 8, 8, 7, 7, 0, 5, 4, 3, 2, 2, 3, 1, 4, 0, 7, 7, 1, 7, 1, 7, 5, 9, 5, 3, 3, 3, 7, 7, 6

Offset: 1

Paul D. Hanna, Dec 03 2015

			z = 0.39318847704964432449725821313890388585483914078866\
28695392932475757877583389749866810976662061018588\
80133300805932263153268090475049426661211424334984\
43584775850655933725091432887705432231407717175953\
33776901692614854937460993931094741172922114373160\
19617637538747813543456758934332723336245738884968...
INFINITE SERIES.
(1) z = 1/4 + 1/4^2 + 3/4^3 + 1/4^4 + 19/4^5 + 25/4^6 + 11/4^8 + 161/4^9 + 227/4^10 + 681/4^11 + 1019/4^12 +...+ A002380(n)/4^n +...
(2) 3 - z = 1/2 + 2/2^2 + 3/2^3 + 5/2^4 + 7/2^5 + 11/2^6 + 17/2^7 + 25/2^8 + 38/2^9 + 57/2^10 + 86/2^11 + 129/2^12 + 194/2^13 + 291/2^14 +...+ A002379(n)/2^n +...
where
3 - z = 2.60681152295035567550274178686109611414516...

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A002379 ([(3/2)^n]), A002380 (3^n mod 2^n), A264919, A264920, A264921, A264922.

z = Sum_{n>=1} (3^n mod 2^n) / 4^n = Sum_{n>=1} A002380(n) / 4^n.

3 - z = Sum_{n>=1} [(3/2)^n] / 2^n = Sum_{n>=1} A002379(n) / 2^n, where [x] denotes the integer floor function of x.

A264920 Decimal expansion of constant z = Sum_{n>=1} {(5/2)^n} * (2/5)^n, where {x} is the fractional part of x.

Original entry on oeis.org

2, 9, 0, 3, 6, 0, 5, 4, 1, 4, 6, 9, 8, 2, 2, 7, 7, 1, 3, 0, 8, 0, 9, 4, 6, 1, 0, 9, 2, 4, 3, 9, 5, 9, 9, 7, 0, 9, 2, 8, 3, 6, 2, 1, 8, 5, 8, 7, 0, 4, 9, 7, 8, 9, 8, 0, 2, 2, 7, 7, 7, 3, 9, 8, 2, 2, 9, 6, 0, 6, 2, 6, 0, 2, 7, 8, 4, 5, 1, 6, 3, 1, 4, 6, 4, 9, 0, 7, 9, 8, 3, 1, 1, 0, 1, 0, 9, 2, 3, 0, 0, 0, 9, 4, 1, 5, 4, 5, 3, 0, 1, 7, 1, 7, 5, 3, 6, 5, 4, 1, 1, 8, 4, 5, 4, 5, 0, 4, 1, 9, 2, 0, 2, 0, 9, 9, 3, 6, 1, 8, 4, 1, 4, 8, 2, 6, 6, 1, 9, 3, 4, 5, 5, 6, 0, 0, 6, 1, 2, 8, 3, 6

Offset: 1

Paul D. Hanna, Dec 03 2015

			z = 0.29036054146982277130809461092439599709283621858704\
97898022777398229606260278451631464907983110109230\
00941545301717536541184545041920209936184148266193\
45560061283623671385240800155746118024134819360960\
04975532010154856783010592635208001206034224504288\
68709199272487962506077008849077084442477839576367\
39796048023434177527610014191473439280227784839521\
53755837195252122355246487888500246551317747557030\
20808674351984895348164983096465432641383339907483\
96807465278185792027753412945335820788419423791331...
INFINITE SERIES.
z = 1/5 + 1/5^2 + 5/5^3 + 1/5^4 + 21/5^5 + 9/5^6 + 45/5^7 + 225/5^8 + 357/5^9 + 761/5^10 + 1757/5^11 + 2641/5^12 + 5013/5^13 + 489/5^14 + 18829/5^15 + 28609/5^16 + 11973/5^17 + 59865/5^18 + 37181/5^19 + 185905/5^20 +...+ A029757(n)/5^n +...

Cf. A029757 (5^n mod 2^n), A264918, A264919, A264921, A264922.

z = Sum_{n>=1} (5^n mod 2^n) / 5^n = Sum_{n>=1} A029757(n) / 5^n.

A264921 Decimal expansion of constant z = Sum_{n>=1} {(4/3)^n} * (3/4)^n, where {x} is the fractional part of x.

Original entry on oeis.org

1, 3, 6, 7, 4, 6, 1, 3, 7, 9, 3, 5, 3, 3, 2, 9, 2, 6, 9, 0, 2, 1, 3, 0, 0, 5, 2, 8, 2, 3, 7, 5, 4, 0, 8, 0, 4, 3, 4, 5, 9, 4, 5, 5, 1, 2, 8, 4, 8, 9, 9, 5, 3, 0, 8, 3, 7, 2, 0, 4, 7, 8, 1, 1, 2, 5, 6, 7, 4, 0, 4, 6, 8, 0, 2, 1, 0, 7, 3, 8, 6, 8, 3, 6, 3, 9, 2, 4, 7, 1, 7, 6, 6, 7, 7, 1, 9, 8, 5, 1, 0, 6, 6, 5, 7, 1, 2, 6, 3, 8, 2, 0, 9, 1, 4, 3, 0, 0, 9, 3, 2, 6, 2, 8, 0, 9, 3, 8, 9, 8, 7, 7, 0, 2, 2, 9, 6, 1, 1, 0, 1, 6, 8, 2, 1, 7, 2, 4, 9, 9, 0, 2, 2, 3, 8, 2, 5, 9, 3, 4, 1, 8, 1, 6, 5, 5, 4, 5, 9, 5, 0, 0, 8, 5, 3, 6, 4, 1, 9, 1, 0, 5, 7, 2, 4, 4, 3, 2, 9

Offset: 1

Paul D. Hanna, Dec 03 2015

			z = 1.3674613793533292690213005282375408043459455128489\
95308372047811256740468021073868363924717667719851\
06657126382091430093262809389877022961101682172499\
02238259341816554595008536419105724432961711520592\
92511101423029805093364719414748469451590148076361\
52981353989027739504422481304813339179550172220838\
78986350689080620566812697277477621308107983782819\
76274774500215875970544025343446657398435575812229\
28979675592867430344641751297842513480112243120370\
37616509374801184872891959991759744341259271254468...
INFINITE SERIES.
z = 1/4 + 7/4^2 + 10/4^3 + 13/4^4 + 52/4^5 + 451/4^6 + 1075/4^7 + 6487/4^8 + 6265/4^9 + 44743/4^10 + 119923/4^11 + 302545/4^12 + 147298/4^13 + 589192/4^14 + 11922706/4^15 + 33341917/4^16 + 4227505/4^17 + 146050183/4^18 + 584200732/4^19 + 1174541461/4^20 +...+ A064629(n)/4^n +...

Cf. A064629 (4^n mod 3^n), A264918, A264919, A264920, A264922.

z = Sum_{n>=1} (4^n mod 3^n) / 4^n = Sum_{n>=1} A064629(n) / 4^n.

A264922 Decimal expansion of constant z = Sum_{n>=1} {2^n/n} * n/2^n, where {x} is the fractional part of x.

Original entry on oeis.org

4, 1, 2, 9, 2, 0, 7, 6, 8, 6, 7, 1, 4, 9, 7, 6, 9, 2, 3, 1, 8, 7, 6, 4, 4, 6, 3, 3, 9, 1, 6, 6, 0, 2, 2, 3, 2, 6, 6, 3, 6, 5, 8, 0, 8, 5, 5, 9, 1, 6, 1, 5, 0, 1, 7, 1, 2, 0, 8, 7, 3, 8, 0, 9, 6, 5, 2, 9, 3, 3, 4, 5, 5, 2, 2, 8, 4, 2, 3, 7, 1, 0, 8, 3, 2, 2, 4, 0, 6, 8, 1, 1, 8, 9, 3, 7, 5, 4, 2, 6, 7, 0, 9, 4, 4, 0, 9, 6, 8, 1, 5, 9, 1, 8, 6, 8, 4, 5, 2, 3, 9, 0, 6, 7, 6, 7, 3, 7, 4, 3, 9, 4, 7, 7, 8, 7, 6, 7, 4, 4, 6, 5, 5, 6, 7, 1, 1, 4, 7, 6, 1, 0, 7, 8, 0, 4, 6, 5, 5, 3, 2, 9

Offset: 0

Paul D. Hanna, Dec 03 2015

			z = 0.41292076867149769231876446339166022326636580855916\
15017120873809652933455228423710832240681189375426\
70944096815918684523906767374394778767446556711476\
10780465532930416417809262367754600548943347721936\
55335089998952017672435611201014919700656911176350\
62372182725523627777491225313970963752168821911399\
67310841379582079241875027376200157722800032983503\
52300500273468914504274753388182612540758874330051\
88409791519634550380640194311077029592977832839103\
92762052659306868595889500273010680885518723259637...
INFINITE SERIES.
z = 0/2 + 0/2^2 + 2/2^3 + 0/2^4 + 2/2^5 + 4/2^6 + 2/2^7 + 0/2^8 + 8/2^9 + 4/2^10 + 2/2^11 + 4/2^12 + 2/2^13 + 4/2^14 + 8/2^15 + 0/2^16 + 2/2^17 + 10/2^18 + 2/2^19 + 16/2^20 + 8/2^21 + 4/2^22 + 2/2^23 + 16/2^24 + 7/2^25 + 4/2^26 + 26/2^27 + 16/2^28 + 2/2^29 + 4/2^30 + 2/2^31 + 0/2^32 + 8/2^33 +...+ A015910(n)/2^n +...

Cf. A015910 (2^n mod n), A264918, A264919, A264920, A264921.

z = Sum_{n>=1} (2^n mod n) / 2^n = Sum_{n>=1} A015910(n) / 2^n.

cp's OEIS Frontend

A264918 Decimal expansion of constant z = Sum_{n>=1} {(3/2)^n} / 2^n, where {x} denotes the fractional part of x.

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A264920 Decimal expansion of constant z = Sum_{n>=1} {(5/2)^n} * (2/5)^n, where {x} is the fractional part of x.

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A264921 Decimal expansion of constant z = Sum_{n>=1} {(4/3)^n} * (3/4)^n, where {x} is the fractional part of x.

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A264922 Decimal expansion of constant z = Sum_{n>=1} {2^n/n} * n/2^n, where {x} is the fractional part of x.

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