A258128
Octagonal numbers (A000567) that are the sum of two consecutive octagonal numbers.
Original entry on oeis.org
5461, 813281, 7272157205, 1083057360705, 9684433559760981, 1442322650052752161, 12896895753596262561301, 1920761265591267733640321, 17174976631595008767000306005, 2557904668044167195987033355105, 22872156829955018609383449248248341
Offset: 1
5461 is in the sequence because Oct(43) = 5461 = 2640 + 2821 = Oct(30) + Oct(31).
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CoefficientList[Series[-x*(x^4 +20*x^3 -1146230*x^2 +807820*x +5461)/((x-1)*(x^2 -1154*x +1)*(x^2 +1154*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1,1331714,-1331714,-1,1},{5461,813281,7272157205,1083057360705,9684433559760981},20] (* Harvey P. Dale, Feb 19 2018 *)
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Vec(-x*(x^4 +20*x^3 -1146230*x^2 +807820*x +5461)/((x-1)*(x^2 -1154*x +1)*(x^2 +1154*x +1)) + O('x^20))
A258130
Octagonal numbers (A000567) that are the sum of ten consecutive octagonal numbers.
Original entry on oeis.org
1045, 1325345, 1910970885, 2755618515265, 3973599987865685, 5729928426883626945, 8262552817966202013445, 11914595433578836419585185, 17180838352667864150839647765, 24774756989951626526674352316385, 35725182398671892783600265200403845
Offset: 1
1045 is in the sequence because Oct(19) = 1045 = 1+8+21+40+65+96+133+176+225+280 = Oct(1) + ... + Oct(10).
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CoefficientList[Series[-95 x (63 x^2 - 1922 x + 11)/((x - 1) (x^2 - 1442 x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1443,-1443,1},{1045,1325345,1910970885},20] (* Harvey P. Dale, Jul 25 2019 *)
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Vec(-95*x*(63*x^2-1922*x+11)/((x-1)*(x^2-1442*x+1)) + O('x^20))
A258131
Octagonal numbers (A000567) that are the sum of eleven consecutive octagonal numbers.
Original entry on oeis.org
49665, 348161, 19701781, 138502485, 7841194625, 55123576321, 3120775694421, 21939044808725, 1242060885120385, 8731684710231681, 494337111502154261, 3475188575627335765, 196744928316972210945, 1383116321414969338241, 78303987133043437737301
Offset: 1
49665 is in the sequence because Oct(129) = 49665 = 3400 + 3605 + 3816 + 4033 + 4256 + 4485 + 4720 + 4961 + 5208 + 5461 + 5720 = Oct(34) + ... + Oct(44).
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CoefficientList[Series[-11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
Select[Total/@Partition[Table[n(3n-2),{n,5*10^6}],11,1],IntegerQ[(Sqrt[1+3#]+1)/3]&] (* Harvey P. Dale, Aug 31 2018 *)
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Vec(-11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)) + O('x^20))
A258132
Octagonal numbers (A000567) that are the sum of fifteen consecutive octagonal numbers.
Original entry on oeis.org
4715040, 8463840, 1122749669280, 2015496399840, 267373851637578960, 479974343542849680, 63672943775553479639280, 114302050117965712710960, 15163202176330482896520455040, 27220118818712616412771202880, 3610995292612020914167620625112640
Offset: 1
4715040 is in the sequence because Oct(1254) = 4715040 = 300833 + 302736 + 304645 + 306560 + 308481 + 310408 + 312341 + 314280 + 316225 + 318176 + 320133 + 322096 + 324065 + 326040 + 328021 = Oct(317) + ... + Oct(331).
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CoefficientList[Series[-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
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Vec(-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)) + O('x^20))
Showing 1-4 of 4 results.