A333009 -id:A333009 - cp's OEIS frontend

A333007 Perfect squares that are sum of two nonzero pentagonal numbers.

36, 121, 196, 400, 625, 1225, 1521, 3481, 5776, 7056, 7225, 8464, 8836, 9025, 9216, 10816, 11025, 11664, 13456, 15129, 15625, 16641, 17689, 18496, 21609, 23409, 24649, 25600, 28561, 30276, 32761, 34596, 35721, 36100, 36864, 37636, 38025, 39204, 40000, 42436, 42849

Offset: 1

Olivier Gérard, Mar 05 2020

			36  = 6^2  = 1  + 35;
121 = 11^2 = 51 + 70;
196 = 14^2 = 51 + 145.

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A000326, A332987, A332988, A333008, A333009.

is(k) = issquare(k) && sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5)>0; \\ Jinyuan Wang, Mar 06 2020

More terms from Jinyuan Wang, Mar 06 2020

A333008 Square roots of perfect squares which are sum of two nonzero pentagonal numbers (A333007).

Original entry on oeis.org

6, 11, 14, 20, 25, 35, 39, 59, 76, 84, 85, 92, 94, 95, 96, 104, 105, 108, 116, 123, 125, 129, 133, 136, 147, 153, 157, 160, 169, 174, 181, 186, 189, 190, 192, 194, 195, 198, 200, 206, 207, 231, 232, 238, 239, 248, 256, 270, 274, 279, 283, 288, 293, 302, 304, 307, 309, 323, 326, 332, 340, 344, 346, 350, 360, 361, 363, 370, 371, 377, 385, 391, 398, 399

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

			a(7) = 39.  a(7)^2 = 1521 = 1 + 1520.

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A333007 (squares), A333009 (successive differences).

Cf. A000326 (pentagonal numbers).

Take[Union[Select[Sqrt[#]&/@(Total/@Tuples[PolygonalNumber[ 5,Range[ 400]],2]),IntegerQ]],80] (* Harvey P. Dale, Apr 22 2022 *)

A332986 Successive differences of index of triangular numbers sum of two pentagonal numbers.

Original entry on oeis.org

1, 4, 6, 18, 2, 7, 1, 4, 1, 2, 4, 5, 5, 3, 1, 2, 2, 5, 3, 3, 4, 3, 1, 10, 1, 1, 8, 3, 4, 7, 5, 12, 10, 8, 9, 7, 3, 5, 3, 6, 3, 10, 11, 1, 2, 13, 9, 6, 3, 2, 1, 2, 3, 3, 2, 2, 6, 13, 9, 4, 1, 11, 1, 3, 8, 9, 1, 4, 6, 3, 8, 3, 5, 2, 3, 2, 5, 1, 14, 9

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

The behavior of this sequence is unexpectedly quite complex.

Peter Kagey, Table of n, a(n) for n = 1..2500

Cf. A000217 (triangular numbers), A000326 (pentagonal numbers), A332987 (sums of two nonzero pentagonal numbers), A333009 (analog for indices of square numbers).

Cf. First differences of A332985.

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A333007 Perfect squares that are sum of two nonzero pentagonal numbers.

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A333008 Square roots of perfect squares which are sum of two nonzero pentagonal numbers (A333007).

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A332986 Successive differences of index of triangular numbers sum of two pentagonal numbers.

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